has a solution in positive integers \(x_{1},\ldots , x_{n}\) . This question was investigated by many authors. For each \(n\ge 3\)  the equation has a solution \((\overline{1}_{n-2}, 2, n)\) . In particular, \(N(n)\ge 1\) , where \(N(n)\)  is the number of solutions. Schinzel showed that there is no other solution for \(n=6\)  or \(n=24\) . He also investigated the existence of rational solutions in the case of \(n=3\)  and proved that for each \(m\)  there are at least \(m\)  triples of integers with the same sum and the same product [1]}. Misiurewicz has shown that \(n=2, 3, 4, 6, 24, 114, 174, 444\)  are the only values of \(n<10^3\)  for which \(N(n)=1\)  [2]} (recall that the value 114 was misprinted as 144 in [2]}). This was later extended by Brown and by Singmaster, Bennett and Dunn to \(n\le 1444000\) . All these results were improved by Weingartner. He proved that \(N(n)>1\)  for \(444<n<10^{11}\)  [4]}. This was possible due to connection with Sophie-Germain primes [5]} (see also [6]}). The question whether there is a value \(n>444\)  such that \(N(n)=1\)  is still open. A nice exposition of the basic results concerning this problem can be found in [7]}. For more on the history of this problem and related investigations see section D24 in [8]}.
